By the work of Langlands summarized in [1]}, a square-integrable automorphic form can be described as residues of Eisenstein series induced from discrete spectrum data on the Levi subgroups of parabolic subgroups. In particular, it is shown by Mœglin-Waldspurger in [2]} that the discrete spectrum of \(GL_n\)  consists of the generalized Speh representations \(\Delta (\tau ,n)\) , which is the irreducible representation generated by the residue at \(\underline{s} = (s_1,\ldots ,s_n) \rightarrow (\frac{n-1}{2},\ldots ,-\frac{n-1}{2})\)  of an Eisenstein series
\(E(\cdot ,\underline{s}): \mathrm {Ind}_{P_{[a^n]}}^{G_{an}}\left(\tau |\cdot |^{s_1}\boxtimes \ldots \boxtimes \tau |\cdot |^{s_n}\right)\rightarrow \mathcal {A}(G_n(k)\backslash G_n(\mathbb {A})).\) 
